The entropy power inequality for quantum systems

TL;DR

This paper extends the classical entropy power inequality to quantum systems, providing proofs that involve new quantum Fisher information concepts and demonstrating the inequality's applicability to bosonic modes at beam splitters.

Contribution

It introduces two quantum generalizations of the entropy power inequality, including new quantum Fisher information and a quantum de Bruijin identity.

Findings

Established quantum entropy power inequalities for bosonic modes.

Developed a quantum de Bruijin identity relating entropy production to Fisher information.

Proved convexity properties of quantum Fisher information in beam splitter contexts.

Abstract

When two independent analog signals, X and Y are added together giving Z=X+Y, the entropy of Z, H(Z), is not a simple function of the entropies H(X) and H(Y), but rather depends on the details of X and Y's distributions. Nevertheless, the entropy power inequality (EPI), which states that exp [2H(Z)] \geq exp[2H(X)] + exp[2H(Y)], gives a very tight restriction on the entropy of Z. This inequality has found many applications in information theory and statistics. The quantum analogue of adding two random variables is the combination of two independent bosonic modes at a beam splitter. The purpose of this work is to give a detailed outline of the proof of two separate generalizations of the entropy power inequality to the quantum regime. Our proofs are similar in spirit to standard classical proofs of the EPI, but some new quantities and ideas are needed in the quantum setting. Specifically, we find a new quantum de Bruijin identity relating entropy production under diffusion to a divergence-based quantum Fisher information. Furthermore, this Fisher information exhibits certain convexity properties in the context of beam splitters.